Abstract
In this paper, we investigate the packing parameters in graphs. By applying the Mantel’s theorem, we give upper bounds on packing and open packing numbers of triangle-free graphs along with characterizing the graphs for which the equalities hold and exhibit sharp Nordhaus–Gaddum type inequalities for packing numbers. We also solve the open problem of characterizing all connected graphs with \(\rho _{o}(G)=n-\omega (G)\) posed in Hamid and Saravanakumar (Discuss Math Graph Theory 35:5–16, 2015).
This is a preview of subscription content, access via your institution.
References
Aouchiche M, Hansen P (2013) A survey of Nordhaus–Gaddum type relations. Discrete Appl Math 161:466–546
Gallant R, Gunther G, Hartnell BL, Rall DF (2010) Limited packing in graphs. Discrete Appl Math 158:1357–1364
Hamid S, Saravanakumar S (2015) Packing parameters in graphs. Discuss Math Graph Theory 35:5–16
Hamid S, Saravanakumar S (2017) On open packing number of graphs. Iran J Math Sci Inform. 12:107–117
Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of domination in graphs. Marcel Dekker, New York
Henning M, Slater PJ (1999) Open packing in graphs. J Combin Math Combin Comput 28:5–18
Mantel W (1907) Problem 28. Wiskundige Opgaven 10:60–61
Mojdeh DA, Samadi B, Hosseini Moghaddam SM (2017) Limited packing vs, tuple domination in graphs. Ars Combinatoria 133:155–161
Samadi B (2016) On the \(k\)-limited packing numbers in graphs. Discrete Optim 22:270–276
West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River
Acknowledgements
We would like to thank the anonymous referee for his/her comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mojdeh, D.A., Samadi, B. Packing parameters in graphs: new bounds and a solution to an open problem. J Comb Optim 38, 739–747 (2019). https://doi.org/10.1007/s10878-019-00410-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-019-00410-4
Keywords
- Packing number
- Open packing number
- Nordhaus–Gaddum inequality
- Open problem
- Triangle-free graph
Mathematics Subject Classification
- 05C69